What is the correct name for a so-called "confidence ellipse"? We often see the terminology "confidence ellipse". This is not correct: in dimension 2, an ellipse is a curve, and the confidence region is the interior of this curve, not the curve. Similarly, in dimension 3, an ellipsoid is a surface, not a volume. And I don't know any name for the interior of an ellipsoid (though this is a ball for an appropriate metric).
What is the correct name, in arbitrary dimension ?


*

*an ellipsoidal confidence region ?

*an elliptic confidence region ?

*an elliptical confidence region ?

*another name ?


In French, my mother tongue, I would hesitate between "ellipsoïdal" and "elliptique". I don't know what is the French for "elliptical".
 A: In common English use we regularly find people using "circle" to refer to the circular disc, not just its boundary, usually without confusion of what was intended. I agree that "ellipsoid" refers to the boundary while that is simply the bounds for the confidence region but I would have little hesitation in saying an ellipsoidal confidence region in spite of it being a bit of a fudging of the terminology. 
If you wanted to be strict you could say something like a confidence region consisting of the interior of an ellipsoid but that seems slightly awkward; the less precise term would be unlikely to be misunderstood in most cases.
In some situations I might say something like "ellipsoidal ball"; while not all that common, it has sometimes been used -- see for example its use on page 652 here (near the middle of the page) -- and would likely be understood.
I'm unaware of a widely accepted single-word term for the interior; while one probably exists, enough people may be unfamiliar with it that it might not be the ideal choice anyway.
